And if you are going for full on real life: The marginal tax rate is piecewise linear (at least in some countries)
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Simon MarkettJun 5 '12 at 12:40

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@SimonMarkett: I think you mean the marginal tax rate is piecewise constant, so that the amount of tax to be payed itself is a piecewise linear function of the income. What you sugfgest would make the tax amount piecewise quadratic, which would be fun (sort of) but is not common.
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Marc van LeeuwenJun 5 '12 at 14:19

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@MarcvanLeeuwen, yes you are right. Since I don't pay tax yet I just vaguely remembered the graph as it was shown to us back in high school and probs confused it. Luckily I am still right strictly speaking, since any constant function is linear :P
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Simon MarkettJun 5 '12 at 14:25

14 Answers
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Something like "buy $5$, and get each one after that at half price" say at a grocery store or clothing store. For example, let $C(x)$ be the cost of the item for the consumer and $p(x)$ be the price (assume it is constant.)

The classic example is friction---say for a block sitting on a rough horizontal plane and subject to a continuously increasing horizontal force. The frictional force rises steadily to match the applied force, and then it drops back a bit to a constant value when the block begins to slide.

Though this, like every example from physics, is of course just an approximation. If you measure precise and quickly enough, the drop of the frictional force is really just a fuzzy transition, governed by randomness (thermal fluctuations).
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leftaroundaboutJun 5 '12 at 17:51

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@leftroundabout: If you measure even more precisely and quickly, you will get only discrete (quantized) transitions in the output of your measuring device.
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John BentinJun 5 '12 at 20:50

Close to absolute zero temperature / in a closed system yes, but not at the conditions where you would usually measure something like friction. Force isn't much good as a quantum-mechanical observable anyway.
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leftaroundaboutJun 5 '12 at 22:46

Is this also called static vs dynamic friction ?
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JackOfAllSep 17 '14 at 12:10

Any disturbed physical system. If you're dealing with circuits you'll often want to solve an equation that involves switches. E.g, letting a capacitor charge for 1 ms and then switching the connection to a closed loop where it discharges. That switch naturally gives a piecewise-defined function. Similar ideas go for physical systems involving collisions, such as a bouncing ball. They come up all the time.

As Wim mentions in the comments, piecewise polynomials are used a fair bit in applications. In designing profiles and shapes for cars, airplanes, and other such devices, one usually uses pieces of Bézier or B-spline curves (or surfaces) during the modeling process, for subsequent machining. In fact, the continuity/smoothness conditions for such curves (usually continuity up to the second derivative) are important here, since during machining, an abrupt change in the curvature can cause the material for the modeling, the mill, or both, to crack (remembering that velocity and acceleration are derivatives of position with respect to time might help to understand why you want smooth curves during machining).

Piecewise constant functions come up all the time in the design and analysis of digital circuits (see square waves, for example). The finite element method is a very widely used technique that approximates solutions of differential equations as piecewise linear functions.